direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.25D4, (C2×C8)⋊6C12, (C2×C24)⋊13C4, C4.3(C6×Q8), C4.Q8⋊13C6, C2.D8⋊13C6, C24.78(C2×C4), C8.16(C2×C12), C12.70(C4⋊C4), C12.92(C2×Q8), (C2×C12).78Q8, (C2×C12).537D4, (C22×C8).13C6, C22.49(C6×D4), C23.29(C3×D4), C6.115(C4○D8), (C22×C24).23C2, C4.26(C22×C12), C42⋊C2.6C6, (C22×C6).128D4, (C2×C12).900C23, (C2×C24).436C22, C12.184(C22×C4), (C22×C12).589C22, C4.21(C3×C4⋊C4), C6.69(C2×C4⋊C4), C2.13(C6×C4⋊C4), C2.2(C3×C4○D8), C4⋊C4.43(C2×C6), (C2×C8).76(C2×C6), (C3×C2.D8)⋊28C2, (C3×C4.Q8)⋊28C2, C22.9(C3×C4⋊C4), (C2×C4).20(C3×Q8), (C2×C6).26(C4⋊C4), (C2×C4).76(C2×C12), (C2×C4).147(C3×D4), (C2×C6).625(C2×D4), (C2×C12).337(C2×C4), (C2×C4).75(C22×C6), (C3×C4⋊C4).364C22, (C22×C4).125(C2×C6), (C3×C42⋊C2).20C2, SmallGroup(192,860)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.25D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 162 in 114 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C42⋊C2, C22×C8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C22×C12, C23.25D4, C3×C4.Q8, C3×C2.D8, C3×C42⋊C2, C22×C24, C3×C23.25D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C4○D8, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C23.25D4, C6×C4⋊C4, C3×C4○D8, C3×C23.25D4
►On 96 points
Generators in S96
(1 39 14)(2 40 15)(3 33 16)(4 34 9)(5 35 10)(6 36 11)(7 37 12)(8 38 13)(17 53 41)(18 54 42)(19 55 43)(20 56 44)(21 49 45)(22 50 46)(23 51 47)(24 52 48)(25 91 71)(26 92 72)(27 93 65)(28 94 66)(29 95 67)(30 96 68)(31 89 69)(32 90 70)(57 78 88)(58 79 81)(59 80 82)(60 73 83)(61 74 84)(62 75 85)(63 76 86)(64 77 87)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 89)(80 90)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(25 64)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 68 19 88)(2 71 20 83)(3 66 21 86)(4 69 22 81)(5 72 23 84)(6 67 24 87)(7 70 17 82)(8 65 18 85)(9 89 46 79)(10 92 47 74)(11 95 48 77)(12 90 41 80)(13 93 42 75)(14 96 43 78)(15 91 44 73)(16 94 45 76)(25 56 60 40)(26 51 61 35)(27 54 62 38)(28 49 63 33)(29 52 64 36)(30 55 57 39)(31 50 58 34)(32 53 59 37)
G:=sub<Sym(96)| (1,39,14)(2,40,15)(3,33,16)(4,34,9)(5,35,10)(6,36,11)(7,37,12)(8,38,13)(17,53,41)(18,54,42)(19,55,43)(20,56,44)(21,49,45)(22,50,46)(23,51,47)(24,52,48)(25,91,71)(26,92,72)(27,93,65)(28,94,66)(29,95,67)(30,96,68)(31,89,69)(32,90,70)(57,78,88)(58,79,81)(59,80,82)(60,73,83)(61,74,84)(62,75,85)(63,76,86)(64,77,87), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,89)(80,90), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,64)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,68,19,88)(2,71,20,83)(3,66,21,86)(4,69,22,81)(5,72,23,84)(6,67,24,87)(7,70,17,82)(8,65,18,85)(9,89,46,79)(10,92,47,74)(11,95,48,77)(12,90,41,80)(13,93,42,75)(14,96,43,78)(15,91,44,73)(16,94,45,76)(25,56,60,40)(26,51,61,35)(27,54,62,38)(28,49,63,33)(29,52,64,36)(30,55,57,39)(31,50,58,34)(32,53,59,37)>;
G:=Group( (1,39,14)(2,40,15)(3,33,16)(4,34,9)(5,35,10)(6,36,11)(7,37,12)(8,38,13)(17,53,41)(18,54,42)(19,55,43)(20,56,44)(21,49,45)(22,50,46)(23,51,47)(24,52,48)(25,91,71)(26,92,72)(27,93,65)(28,94,66)(29,95,67)(30,96,68)(31,89,69)(32,90,70)(57,78,88)(58,79,81)(59,80,82)(60,73,83)(61,74,84)(62,75,85)(63,76,86)(64,77,87), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,89)(80,90), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(25,64)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,68,19,88)(2,71,20,83)(3,66,21,86)(4,69,22,81)(5,72,23,84)(6,67,24,87)(7,70,17,82)(8,65,18,85)(9,89,46,79)(10,92,47,74)(11,95,48,77)(12,90,41,80)(13,93,42,75)(14,96,43,78)(15,91,44,73)(16,94,45,76)(25,56,60,40)(26,51,61,35)(27,54,62,38)(28,49,63,33)(29,52,64,36)(30,55,57,39)(31,50,58,34)(32,53,59,37) );
G=PermutationGroup([[(1,39,14),(2,40,15),(3,33,16),(4,34,9),(5,35,10),(6,36,11),(7,37,12),(8,38,13),(17,53,41),(18,54,42),(19,55,43),(20,56,44),(21,49,45),(22,50,46),(23,51,47),(24,52,48),(25,91,71),(26,92,72),(27,93,65),(28,94,66),(29,95,67),(30,96,68),(31,89,69),(32,90,70),(57,78,88),(58,79,81),(59,80,82),(60,73,83),(61,74,84),(62,75,85),(63,76,86),(64,77,87)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52),(65,85),(66,86),(67,87),(68,88),(69,81),(70,82),(71,83),(72,84),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,89),(80,90)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(25,64),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,95),(74,96),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,68,19,88),(2,71,20,83),(3,66,21,86),(4,69,22,81),(5,72,23,84),(6,67,24,87),(7,70,17,82),(8,65,18,85),(9,89,46,79),(10,92,47,74),(11,95,48,77),(12,90,41,80),(13,93,42,75),(14,96,43,78),(15,91,44,73),(16,94,45,76),(25,56,60,40),(26,51,61,35),(27,54,62,38),(28,49,63,33),(29,52,64,36),(30,55,57,39),(31,50,58,34),(32,53,59,37)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12AB | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D4 | Q8 | D4 | C3×D4 | C3×Q8 | C3×D4 | C4○D8 | C3×C4○D8 |
| kernel | C3×C23.25D4 | C3×C4.Q8 | C3×C2.D8 | C3×C42⋊C2 | C22×C24 | C23.25D4 | C2×C24 | C4.Q8 | C2.D8 | C42⋊C2 | C22×C8 | C2×C8 | C2×C12 | C2×C12 | C22×C6 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 1 | 2 | 1 | 2 | 4 | 2 | 8 | 16 |
Matrix representation of C3×C23.25D4 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 22 |
| 46 | 0 | 0 |
| 0 | 0 | 22 |
| 0 | 10 | 0 |
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[1,0,0,0,72,0,0,0,1],[72,0,0,0,72,0,0,0,72],[1,0,0,0,72,0,0,0,72],[72,0,0,0,63,0,0,0,22],[46,0,0,0,0,10,0,22,0] >;
C3×C23.25D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{25}D_4 % in TeX
G:=Group("C3xC2^3.25D4"); // GroupNames label
G:=SmallGroup(192,860);
// by ID
G=gap.SmallGroup(192,860);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,520,4204,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^4=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations